Question

If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two unit vectors and $\theta$ is the angle between them, then $\frac{1}{2}|\overrightarrow{x}-\overrightarrow{y}|$ is equal to

• A. 0
• B. $\frac{x}{2}$
• C. $cos\frac{\theta }{2}$
• D. $sin\frac{\theta }{2}$

Answer 1

Answer: (D)

$\frac{1}{4}{\left|\vec{x}-\vec{y}\right|}^2$
$=\frac{1}{4}{\left(\bar{x}-\bar{y}\right)}^2=\frac{1}{4}\left({\vec{x}}^2+{\vec{y}}^2-2\vec{x}\cdot \vec{y}\right)$
$=\frac{1}{4}\left(1+1-2cos\theta \right)$
$=\frac{2}{4}\left(1-cos\theta \right)=si{n}^2\frac{\theta }{2}$
$\therefore \frac{1}{2}\left|\vec{x}-\vec{y}\right|=sin\frac{\theta }{2}$